2.2. LAPLACE’S EQUATION 29
c. Local estimates for harmonic functions. Now we employ the mean-
value formulas to derive careful estimates on the various partial derivatives
of a harmonic function. The precise structure of these estimates will be
needed below, when we prove analyticity.
THEOREM 7 (Estimates on derivatives). Assume u is harmonic in U.
for each ball B(x0,r) ⊂ U and each multiindex α of order |α| = k.
(19) C0 =
, Ck =
(k = 1,... ).
Proof. 1. We establish (18), (19) by induction on k, the case k = 0 being
immediate from the mean-value formula (16). For k = 1, we note upon
diﬀerentiating Laplace’s equation that uxi (i = 1,...,n) is harmonic. Con-
|uxi (x0)| = −
Now if x ∈
∂B(x0,r/2), then B(x, r/2) ⊂ B(x0,r) ⊂ U, and so
by (18), (19) for k = 0. Combining the inequalities above, we deduce
if |α| = 1. This veriﬁes (18), (19) for k = 1.
2. Assume now k ≥ 2 and (18), (19) are valid for all balls in U and each
multiindex of order less than or equal to k − 1. Fix B(x0,r) ⊂ U and let α